Benjamin (1962) introduced the idea that a given ‘primary’ swirling flow, with cylindrical stream surfaces, may have associated with it ‘conjugate flows’, also swirling and cylindrical, which in a certain sense are equivalent to the primary one. He deduced that, in cases where such conjugate flows exist and where the primary flow cannot support standing waves of small amplitude, the conjugate flow nearest the primary one (a) can support such waves, and (b) has a ‘flow force’ greater than that of the primary flow. In the present paper these two results are proved rigorously by a method which differs from Benjamin's.

Experimental results are given for various statistical properties of the fluctuating wall-pressure field associated with a subsonic turbulent equilibrium boundary layer, developed on a smooth wind tunnel wall after natural transition from laminar to turbulent flow.

The statistical quantities of the wall-pressure field investigated were root-mean-square pressure, frequency power spectrum and space-time correlations. Space-time correlation measurements were made in both broad and narrow frequency bands. The experiments were made at flow Mach numbers of 0·3 and 0·5 and covered a Reynolds number range of about 5 to 1.

The main conclusion to which the measurements lead is that the wall-pressure field has a structure produced by contributions from pressure sources in the boundary layer with a wide range of convection velocities, and comprises two families of convected wave-number components. One family is of high wave-number components and is associated with turbulent motion in the constant stress layer; the components are longitudinally coherent for times proportional to the times taken for them to be convected distances equal to their wavelengths and laterally coherent over distances proportional to their wavelengths. The other family comprises components of wavelength greater than about twice the boundary-layer thickness, which lose coherence as a group more or less independently of wavelength and are associated with large-scale eddy motion in the boundary layer, outside the constant stress layer. The evolution of the pressure field is discussed in terms of these two wave-number families.

Asymptotic equations of long waves in a compressible medium of infinite height with arbitrary density and wind profiles, including cross-wind, are systematically derived to provide the mathematical description of certain pressure-front phenomena observed in the atmosphere. It is assumed that the final state of the wave is a plane wave moving with nearly constant velocity, even though its velocity field has a component transverse to the direction of propagation. The coefficients of the asymptotic long-wave equations are found to depend on the equilibrium profiles of the density and of the velocity component in the direction of propagation of the wave, but not on the profile of the transverse velocity. Any horizontal direction of propagation is found to be possible, even for waves of permanent type.

Dynamically passive transfer of heat across a layer of liquid supporting a progressive, periodic surface wave is approached from a Lagrangian viewpoint. The model layer considered is the region between two constant pressure surfaces of a Gerstner wave and the thermal boundary conditions are that the average temperature of any surface particle remains constant and that there is horizontal homogeneity of the average temperature field.

It is shown that fluctuations in the temperature of any particle are negligibly small for ordinary liquids and a uniformly valid approximation to the average temperature of each particle is presented.

The extent to which the flux of heat through the layer is augmented is computed for typical cases and it is shown to be at most doubled. Indication is given of extensions of the method to other kinds of progressive waves and to situations in which the boundary conditions are unsteady and spatially inhomogeneous.

The velocity of a large gas bubble rising along the axis of a cylindrical container filled with liquid, is derived to a first approximation from a simple flow model of the system. The resulting expressions involve ratios of certain infinite series, which are evaluated for a few cases, and the theory is then seen to agree with previously accepted results at its asymptotes. Experiments are performed which confirm the first approximation and which allow the theory to be recast as a semi-empirical theory relating bubble volume with velocity. In this form it agrees with the results of Uno & Kintner (1956), but indicates that the relation between volume and curvature which they employed was in error.

The diffraction of gravity waves at a discontinuity in depth is described by a scattering matrix that relates the asymptotic, plane-wave fields (each of which may contain waves travelling towards and away from the discontinuity) on the two sides of the discontinuity. Plane-wave and variational approximations for the elements of this scattering matrix are developed. These approximate results are tested by comparison with the more accurate results obtained by Newman for an infinite step. The plane-wave approximation to the magnitude of the transmission coefficient is within 5% of Newman's result for all wavelengths, but the corresponding approximation to the reflexion coefficient is satisfactory only for rather long wavelengths. The variational approximations to the complex transmission and reflexion coefficients agree with Newman's results, within the accuracy with which his graphs can be read, for all wavelengths. The variational approximations also are used to determine the effects of trapped modes on the resonant width of a shelf that terminates at a vertical cliff.

A stably stratified shear flow has the velocity profile V(1 - e−y/h) and the density profile log(ρ/ρ) = σ(1 − e−y/h) in 0 < y < ∞ and is bounded by the rigid plane y = 0. It is proved that small disturbances with respect to this basic flow are stable for all wavelengths and Richardson numbers. The eigenvalues for neutral disturbances (internal gravity waves) are enumerated. The results are applicable to the atmosphere and to an infinitely deep ocean.

When a horizontal layer of viscous liquid with an adsorption film of varying concentration as its upper boundary is set in motion by a steady translation of its lower boundary, plane Couette flow with zero surface velocity is possible. In this paper the stability of this flow is considered, and it is found that the liquid layer can be unstable for long waves. The instability found for this flow, however, exists also for other laminar flows with an adsorbed film, and plane Couette flow has been chosen only as a simple means of demonstration.

Careful plume-rise observations by photographic means have been made on smoke plumes from the Lakeview Generating Station (Ontario) and compared with some existing theoretical formulae. Supporting data in considerable detail on stack parameters were available.

A linearly rising portion of the plume was found beyond a specific transition point in a neutral atmosphere. Also, the previously established simple power law $z \sim F^{\frac{1}{3}} U^{-1} x^{\frac{2}{3}}$, was found to agree quite well with observation up to the transition point.

In an unstable atmosphere the plume was sometimes above and sometimes below a corresponding plume in neutral conditions, under the opposing influences of increased dilution and the direct effect of instability in promoting plume rise.

Results are presented of a preliminary numerical investigation into a three-dimensional laminar boundary layer. It is assumed that the flow is over a developable surface and the boundary conditions at the outer edge of the layer are chosen to be u = U0 + xU1, v = yV1. This choice enables the governing equations to be written in terms of two, and not three, independent variables, viz. x and z. However, the three-dimensionality of the problem gives rise to a coupling of the equations which, not unnaturally, is still present after the elimination of y.

For appropriate values of U1 and V1 it is found possible to integrate the equations approximately from the ‘birth’ of the boundary layer (x = 0) right up to a saddle point of attachment. Calculations have already been made for flow at such attachment points and the comparison of the present results with them is extremely good.

The effect of the transverse motion of a cylinder on its natural vortex wake is examined at various driving frequencies giving special attention to the change in wake geometry along the span. Conditions are established for which the vortex wake frequency is controlled by the driving frequency of the cylinder.

This paper describes a very general method for determining the steady two-dimensional potential flow about one or more bodies of arbitrary shape operating at arbitrary Froude number near a free surface. The boundary condition of zero velocity (solid wall) or prescribed velocity (suction or blowing) normal to the body surface is satisfied exactly, and the boundary condition of constant pressure on the free surface is satisfied using the classic small-wave approximation. Calculations made by the present method are compared with analytic results, other theoretical calculations and experimental data. Examples for which no comparison exists are also presented to illustrate the capability of the method.

A study is made of hydromagnetic oscillations in a rotating fluid sphere. The basic state is chosen as a uniform current parallel to the axis of rotation. It is found that the non-dissipative normal modes are described by a modified form of the Poincaré eigenvalue problem. For small rotation rates, the lowest non-axisymmetric modes are unstable. For rotation rates of geophysical interest all normal modes are stable. The introduction of ohmic dissipation leads to a hydro-magnetic boundary-layer problem. Solutions for the boundary layer are outlined indicating its role in altering the free periods, damping the oscillations and producing external magnetic fields. Dispersion relations are derived which establish that the zonal phase velocities of both ‘fast’ hydrodynamic and ‘slow’ hydro-magnetic waves can be of either sign. Observations of the secular variations of the earth's magnetic field indicate motion primarily towards the west. A mechanism for the selective excitation of the observed motion is discussed.

Axially symmetric potential flow about an axially symmetric rigid body is considered. The potential due to the body is represented as a superposition of potentials of point sources distributed along a segment of the axis inside the body. The source strength distribution satisfies a linear integral equation. A complete uniform asymptotic expansion of its solution is obtained with respect to the slenderness ratio ε½, which is the maximum radius of the body divided by its length. The expansion contains integral powers of ε multiplied by powers of log ε. From it expansions of the potential, the virtual mass and the dipole moment of the body are obtained. The flow about the body in the presence of an axially symmetric stationary obstacle is also determined. The method of analysis involves a technique for the asymptotic solution of integral equations.

An extension of numerical methods described in previous papers is used to analyse the time-dependent rotationally-symmetric motion of an incompressible viscous fluid contained between two concentric spheres having a common axis of rotation. The motion is governed by a pair of coupled non-linear partial differential equations in three independent variables, with singular end conditions. The computational process is described, and numerical solutions are presented for cases in which one (or both) of the spheres is given an impulsive change in angular velocity-starting from a state of either rest or uniform rotation. Reynolds numbers lie in the range 10-1500.

An axisymmetric viscous-transonic equation is presented. A nozzle-type similarity solution of this equation has been found, which describes the initial stages in the development of shock waves downstream of a converging-diverging nozzle throat. This solution is an extension of a two-dimensional solution found previously (Sichel 1966). By an appropriate choice of an arbitrary scaling constant solutions were found such that there is essentially a weak normal shock near the axis with effects of wall and shock-wave curvature occurring only at a sufficiently large radius. The upstream and downstream asymptotic behaviour of these viscous-transonic nozzle solutions has been investigated.

When thin films of liquid are produced by rolling it is commonly observed that the liquid film is not uniform but has regularly spaced thick lines or ribs. This phenomenon has been studied experimentally and it was found that the onset of ribbing was governed by the relation $\eta S/\gamma = 10.3 (g/R)^{\frac{3}{4}}$. At higher values of the parameter η/Sγ the number of ribs per centimetre increased sharply. At values of ηS/γ > 2·4 the rib number became independent of this parameter and varied solely with the geometrical parameters of the system, i.e. roller radius and gap.

The possible occurrence of a viscous region near a surface from which there is rapid efflux of gas, accompanied by large heat transfer, is postulated and investigated theoretically. Such a viscous region, denoted as a boundary shock wave, may occur in the case of a large high-speed meteor entering the earth's atmosphere, when a very high rate of vaporization induces translational non-equilibrium. The conditions across a boundary shock wave and its structure are calculated from the appropriate macroscopic equations reduced to closed-form expressions under the restrictions of a perfect gas, flowing at constant total enthalpy.

This paper shows the existence of a three-dimensional solution of the boundary-layer equations of viscous incompressible flow in the immediate neighbourhood of a two-dimensional stagnation point of attachment. The numerical solution has been obtained.

A simple wall-turbulence interaction has been studied experimentally. In the idealized model an infinite flat plate is suddenly inserted into a pre-existing field of homogeneous isotropic turbulence, and subsequent changes in the turbulence field examined. The experiment involved passing grid-produced turbulence over a wall moving at the mean speed. Mean velocity gradients vanish in both the model and experiment, and hence production of new turbulence is absent. This allowed the inhibiting effects of the wall to be studied separately. The growth of the ‘inhomogeneity layer’ into the impressed turbulence field and other statistical features of the turbulence were measured.